NMaximize is not converging to a solution












3












$begingroup$


I am trying to use NMaximize to find the maximum value of a variable that satisfies the given constraints. Since the constraints aren't straightforward, I am using the function.



I can see the constraints are such that the value is bounded but I get the below warning messages:




NMaximize::cvmit: Failed to converge to the requested accuracy or
precision within 100000 iterations.



NMaximize::cvdiv: Failed to
converge to a solution. The function may be unbounded.




The constraint and the way I am using the function is as below:



    constraint = (x | y) [Element] 

  Integers && ((x == 0 && 1. <= y <= 12720.) || (1. <= x <= 10712. && 

     0 <= y < 

      2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) + 

       2.8484*10^-43 Sqrt[

        4.98614*10^92 + 4.65469*10^88 x - 

         3.63201*10^84 x^2]) || (10713. <= x <= 19762. && 

     2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) - 

       2.8484*10^-43 Sqrt[

        4.98614*10^92 + 4.65469*10^88 x - 3.63201*10^84 x^2] < y < 

      2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) + 

       2.8484*10^-43 Sqrt[

        4.98614*10^92 + 4.65469*10^88 x - 3.63201*10^84 x^2]))





maxX =  

 NMaximize[{x, constraint}, {x, y}, MaxIterations -> 100000]


I have increased the MaxIterations from 100 to 100000 but it doesn't seem to converge. I am not sure if increasing the MaxIterations is the solution. Can you please guide me with this?






functions maximum







share|improve this question









$endgroup$












  • $begingroup$
    Could try maximizing over individual regions of the piecewise set-up. But the machine precision values will make validation of inequalities kind of iffy.
    $endgroup$
    – Daniel Lichtblau
    11 hours ago






  • 1




    $begingroup$
    I'm not seeing what $y$ has to do with this. Wouldn't the maximum value of $x$ be 19762? constraint /. x -> 19762 results in y [Element] Integers && 7229.16 < y < 7344.29 and constraint /. x -> 19763 results in False.
    $endgroup$
    – JimB
    11 hours ago












  • $begingroup$
    @JimB, I think for x, y isn't needed. Thanks for pointing this out. But if I am trying to maximize y, I need to maximize over both the variables since y is an expression of x, right?
    $endgroup$
    – gaganso
    11 hours ago










  • $begingroup$
    Yes, if that's what you want. The general solution appears to be $x = 19762$ and $7230leq y leq 7344$. So to maximize $y$ you'd choose $7344$.
    $endgroup$
    – JimB
    10 hours ago








  • 1




    $begingroup$
    OK. I was assuming that you were conditioning on the maximum value of $x$.
    $endgroup$
    – JimB
    10 hours ago
















3












$begingroup$


I am trying to use NMaximize to find the maximum value of a variable that satisfies the given constraints. Since the constraints aren't straightforward, I am using the function.



I can see the constraints are such that the value is bounded but I get the below warning messages:




NMaximize::cvmit: Failed to converge to the requested accuracy or
precision within 100000 iterations.



NMaximize::cvdiv: Failed to
converge to a solution. The function may be unbounded.




The constraint and the way I am using the function is as below:



    constraint = (x | y) [Element] 

  Integers && ((x == 0 && 1. <= y <= 12720.) || (1. <= x <= 10712. && 

     0 <= y < 

      2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) + 

       2.8484*10^-43 Sqrt[

        4.98614*10^92 + 4.65469*10^88 x - 

         3.63201*10^84 x^2]) || (10713. <= x <= 19762. && 

     2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) - 

       2.8484*10^-43 Sqrt[

        4.98614*10^92 + 4.65469*10^88 x - 3.63201*10^84 x^2] < y < 

      2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) + 

       2.8484*10^-43 Sqrt[

        4.98614*10^92 + 4.65469*10^88 x - 3.63201*10^84 x^2]))





maxX =  

 NMaximize[{x, constraint}, {x, y}, MaxIterations -> 100000]


I have increased the MaxIterations from 100 to 100000 but it doesn't seem to converge. I am not sure if increasing the MaxIterations is the solution. Can you please guide me with this?






functions maximum







share|improve this question









$endgroup$












  • $begingroup$
    Could try maximizing over individual regions of the piecewise set-up. But the machine precision values will make validation of inequalities kind of iffy.
    $endgroup$
    – Daniel Lichtblau
    11 hours ago






  • 1




    $begingroup$
    I'm not seeing what $y$ has to do with this. Wouldn't the maximum value of $x$ be 19762? constraint /. x -> 19762 results in y [Element] Integers && 7229.16 < y < 7344.29 and constraint /. x -> 19763 results in False.
    $endgroup$
    – JimB
    11 hours ago












  • $begingroup$
    @JimB, I think for x, y isn't needed. Thanks for pointing this out. But if I am trying to maximize y, I need to maximize over both the variables since y is an expression of x, right?
    $endgroup$
    – gaganso
    11 hours ago










  • $begingroup$
    Yes, if that's what you want. The general solution appears to be $x = 19762$ and $7230leq y leq 7344$. So to maximize $y$ you'd choose $7344$.
    $endgroup$
    – JimB
    10 hours ago








  • 1




    $begingroup$
    OK. I was assuming that you were conditioning on the maximum value of $x$.
    $endgroup$
    – JimB
    10 hours ago














3












3








3





$begingroup$


I am trying to use NMaximize to find the maximum value of a variable that satisfies the given constraints. Since the constraints aren't straightforward, I am using the function.



I can see the constraints are such that the value is bounded but I get the below warning messages:




NMaximize::cvmit: Failed to converge to the requested accuracy or
precision within 100000 iterations.



NMaximize::cvdiv: Failed to
converge to a solution. The function may be unbounded.




The constraint and the way I am using the function is as below:



    constraint = (x | y) [Element] 

  Integers && ((x == 0 && 1. <= y <= 12720.) || (1. <= x <= 10712. && 

     0 <= y < 

      2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) + 

       2.8484*10^-43 Sqrt[

        4.98614*10^92 + 4.65469*10^88 x - 

         3.63201*10^84 x^2]) || (10713. <= x <= 19762. && 

     2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) - 

       2.8484*10^-43 Sqrt[

        4.98614*10^92 + 4.65469*10^88 x - 3.63201*10^84 x^2] < y < 

      2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) + 

       2.8484*10^-43 Sqrt[

        4.98614*10^92 + 4.65469*10^88 x - 3.63201*10^84 x^2]))





maxX =  

 NMaximize[{x, constraint}, {x, y}, MaxIterations -> 100000]


I have increased the MaxIterations from 100 to 100000 but it doesn't seem to converge. I am not sure if increasing the MaxIterations is the solution. Can you please guide me with this?






functions maximum







share|improve this question









$endgroup$




I am trying to use NMaximize to find the maximum value of a variable that satisfies the given constraints. Since the constraints aren't straightforward, I am using the function.



I can see the constraints are such that the value is bounded but I get the below warning messages:




NMaximize::cvmit: Failed to converge to the requested accuracy or
precision within 100000 iterations.



NMaximize::cvdiv: Failed to
converge to a solution. The function may be unbounded.




The constraint and the way I am using the function is as below:



    constraint = (x | y) [Element] 

  Integers && ((x == 0 && 1. <= y <= 12720.) || (1. <= x <= 10712. && 

     0 <= y < 

      2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) + 

       2.8484*10^-43 Sqrt[

        4.98614*10^92 + 4.65469*10^88 x - 

         3.63201*10^84 x^2]) || (10713. <= x <= 19762. && 

     2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) - 

       2.8484*10^-43 Sqrt[

        4.98614*10^92 + 4.65469*10^88 x - 3.63201*10^84 x^2] < y < 

      2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) + 

       2.8484*10^-43 Sqrt[

        4.98614*10^92 + 4.65469*10^88 x - 3.63201*10^84 x^2]))





maxX =  

 NMaximize[{x, constraint}, {x, y}, MaxIterations -> 100000]


I have increased the MaxIterations from 100 to 100000 but it doesn't seem to converge. I am not sure if increasing the MaxIterations is the solution. Can you please guide me with this?






functions maximum




functions maximum






share|improve this question













share|improve this question











share|improve this question




share|improve this question










asked 12 hours ago









gagansogaganso

1528




1528












  • $begingroup$
    Could try maximizing over individual regions of the piecewise set-up. But the machine precision values will make validation of inequalities kind of iffy.
    $endgroup$
    – Daniel Lichtblau
    11 hours ago






  • 1




    $begingroup$
    I'm not seeing what $y$ has to do with this. Wouldn't the maximum value of $x$ be 19762? constraint /. x -> 19762 results in y [Element] Integers && 7229.16 < y < 7344.29 and constraint /. x -> 19763 results in False.
    $endgroup$
    – JimB
    11 hours ago












  • $begingroup$
    @JimB, I think for x, y isn't needed. Thanks for pointing this out. But if I am trying to maximize y, I need to maximize over both the variables since y is an expression of x, right?
    $endgroup$
    – gaganso
    11 hours ago










  • $begingroup$
    Yes, if that's what you want. The general solution appears to be $x = 19762$ and $7230leq y leq 7344$. So to maximize $y$ you'd choose $7344$.
    $endgroup$
    – JimB
    10 hours ago








  • 1




    $begingroup$
    OK. I was assuming that you were conditioning on the maximum value of $x$.
    $endgroup$
    – JimB
    10 hours ago


















  • $begingroup$
    Could try maximizing over individual regions of the piecewise set-up. But the machine precision values will make validation of inequalities kind of iffy.
    $endgroup$
    – Daniel Lichtblau
    11 hours ago






  • 1




    $begingroup$
    I'm not seeing what $y$ has to do with this. Wouldn't the maximum value of $x$ be 19762? constraint /. x -> 19762 results in y [Element] Integers && 7229.16 < y < 7344.29 and constraint /. x -> 19763 results in False.
    $endgroup$
    – JimB
    11 hours ago












  • $begingroup$
    @JimB, I think for x, y isn't needed. Thanks for pointing this out. But if I am trying to maximize y, I need to maximize over both the variables since y is an expression of x, right?
    $endgroup$
    – gaganso
    11 hours ago










  • $begingroup$
    Yes, if that's what you want. The general solution appears to be $x = 19762$ and $7230leq y leq 7344$. So to maximize $y$ you'd choose $7344$.
    $endgroup$
    – JimB
    10 hours ago








  • 1




    $begingroup$
    OK. I was assuming that you were conditioning on the maximum value of $x$.
    $endgroup$
    – JimB
    10 hours ago
















$begingroup$
Could try maximizing over individual regions of the piecewise set-up. But the machine precision values will make validation of inequalities kind of iffy.
$endgroup$
– Daniel Lichtblau
11 hours ago




$begingroup$
Could try maximizing over individual regions of the piecewise set-up. But the machine precision values will make validation of inequalities kind of iffy.
$endgroup$
– Daniel Lichtblau
11 hours ago




1




1




$begingroup$
I'm not seeing what $y$ has to do with this. Wouldn't the maximum value of $x$ be 19762? constraint /. x -> 19762 results in y [Element] Integers && 7229.16 < y < 7344.29 and constraint /. x -> 19763 results in False.
$endgroup$
– JimB
11 hours ago






$begingroup$
I'm not seeing what $y$ has to do with this. Wouldn't the maximum value of $x$ be 19762? constraint /. x -> 19762 results in y [Element] Integers && 7229.16 < y < 7344.29 and constraint /. x -> 19763 results in False.
$endgroup$
– JimB
11 hours ago














$begingroup$
@JimB, I think for x, y isn't needed. Thanks for pointing this out. But if I am trying to maximize y, I need to maximize over both the variables since y is an expression of x, right?
$endgroup$
– gaganso
11 hours ago




$begingroup$
@JimB, I think for x, y isn't needed. Thanks for pointing this out. But if I am trying to maximize y, I need to maximize over both the variables since y is an expression of x, right?
$endgroup$
– gaganso
11 hours ago












$begingroup$
Yes, if that's what you want. The general solution appears to be $x = 19762$ and $7230leq y leq 7344$. So to maximize $y$ you'd choose $7344$.
$endgroup$
– JimB
10 hours ago






$begingroup$
Yes, if that's what you want. The general solution appears to be $x = 19762$ and $7230leq y leq 7344$. So to maximize $y$ you'd choose $7344$.
$endgroup$
– JimB
10 hours ago






1




1




$begingroup$
OK. I was assuming that you were conditioning on the maximum value of $x$.
$endgroup$
– JimB
10 hours ago




$begingroup$
OK. I was assuming that you were conditioning on the maximum value of $x$.
$endgroup$
– JimB
10 hours ago










2 Answers
2






active

oldest

votes


















5












$begingroup$

Rationalize the constraint:



constraint2 = ((x == 0 && 1. <= y <= 12720.) || (1. <= x <= 10712. && 

        0 <= y < 2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) + 

          2.8484*10^-1 Sqrt[

            4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2]) || (10713. <= x <= 

         19762. && 

        2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) - 

          2.8484*10^-1 Sqrt[4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2] < y <

          2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) + 

          2.8484*10^-1 Sqrt[4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2])) // 

    Rationalize[#, 0] & // Simplify;


With the Rationalized constraint you can use Maximize:



maxX = Maximize[{x, constraint2}, {x, y}]



(* {19762, {x -> 19762, y -> 7287}} *)



constraint2 /. maxX[[2]]



(* True *)


EDIT: To find maximum y



(maxY = Maximize[{y, constraint2}, {x, y}]) // N


enter image description here



To plot the region defined by the constraint:



reg = ImplicitRegion[constraint2, {x, y}];



Region[reg,

 Frame -> True,

 FrameLabel -> (Style[#, 12, Bold] & /@ {x, y}),

 Epilog -> {Red,

   AbsolutePointSize[3],

   Point[{x, y} /. maxX[[2]]],

   Point[{x, y} /. maxY[[2]]]}]


enter image description here






share|improve this answer











$endgroup$





















    3












    $begingroup$

    You have numbers spread a wide range of magnitudes for no good reason. This range is probably too wide for machine precision arithmetic. Also telling NMinimize explicitly that this an integer optimization problem seems to help. Try this:



    constraint2 = ((x == 0 && 1. <= y <= 12720.) || (1. <= x <= 10712. && 
    
      0 <= y < 
    
       2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) + 
    
        2.8484*10^-1 Sqrt[
    
          4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2]) || (10713. <= 
    
       x <= 19762. && 
    
      2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) - 
    
        2.8484*10^-1 Sqrt[
    
          4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2] < y < 
    
       2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) + 
    
        2.8484*10^-1 Sqrt[
    
          4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2])) // Expand
    

    
maxX = NMaximize[{x, constraint2}, {x, y}, Integers, 
    
  MaxIterations -> 10000]



    {19762., {x -> 19762, y -> 7311}}




    And with your definition of constraint:



    constraint /. maxX[[2]]



    True







    share|improve this answer











    $endgroup$













    • $begingroup$
      But constraint /. x -> 19762 /. y -> 8647 results in False?
      $endgroup$
      – JimB
      11 hours ago










    • $begingroup$
      @JimB D'oh. Yeah, I did the simplification wrong. -.- Thanks for pointing that out.
      $endgroup$
      – Henrik Schumacher
      11 hours ago












    • $begingroup$
      @HenrikSchumacher, thank you for this. This works for x but when I try to find the maximum y similarly, I still get the same message - NMaximize[{y, res}, {x, y}, Integers, MaxIterations -> 100000]. Output: NMaximize::cvdiv: Failed to converge to a solution. The function may be unbounded.
      $endgroup$
      – gaganso
      11 hours ago












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    2 Answers
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    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    5












    $begingroup$

    Rationalize the constraint:



    constraint2 = ((x == 0 && 1. <= y <= 12720.) || (1. <= x <= 10712. && 
    
        0 <= y < 2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) + 
    
          2.8484*10^-1 Sqrt[
    
            4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2]) || (10713. <= x <= 
    
         19762. && 
    
        2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) - 
    
          2.8484*10^-1 Sqrt[4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2] < y <
    
          2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) + 
    
          2.8484*10^-1 Sqrt[4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2])) // 
    
    Rationalize[#, 0] & // Simplify;


    With the Rationalized constraint you can use Maximize:



    maxX = Maximize[{x, constraint2}, {x, y}]
    

    
(* {19762, {x -> 19762, y -> 7287}} *)
    

    
constraint2 /. maxX[[2]]
    

    
(* True *)


    EDIT: To find maximum y



    (maxY = Maximize[{y, constraint2}, {x, y}]) // N


    enter image description here



    To plot the region defined by the constraint:



    reg = ImplicitRegion[constraint2, {x, y}];
    

    
Region[reg,
    
 Frame -> True,
    
 FrameLabel -> (Style[#, 12, Bold] & /@ {x, y}),
    
 Epilog -> {Red,
    
   AbsolutePointSize[3],
    
   Point[{x, y} /. maxX[[2]]],
    
   Point[{x, y} /. maxY[[2]]]}]


    enter image description here






    share|improve this answer











    $endgroup$


















      5












      $begingroup$

      Rationalize the constraint:



      constraint2 = ((x == 0 && 1. <= y <= 12720.) || (1. <= x <= 10712. && 
      
        0 <= y < 2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) + 
      
          2.8484*10^-1 Sqrt[
      
            4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2]) || (10713. <= x <= 
      
         19762. && 
      
        2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) - 
      
          2.8484*10^-1 Sqrt[4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2] < y <
      
          2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) + 
      
          2.8484*10^-1 Sqrt[4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2])) // 
      
    Rationalize[#, 0] & // Simplify;


      With the Rationalized constraint you can use Maximize:



      maxX = Maximize[{x, constraint2}, {x, y}]
      

      
(* {19762, {x -> 19762, y -> 7287}} *)
      

      
constraint2 /. maxX[[2]]
      

      
(* True *)


      EDIT: To find maximum y



      (maxY = Maximize[{y, constraint2}, {x, y}]) // N


      enter image description here



      To plot the region defined by the constraint:



      reg = ImplicitRegion[constraint2, {x, y}];
      

      
Region[reg,
      
 Frame -> True,
      
 FrameLabel -> (Style[#, 12, Bold] & /@ {x, y}),
      
 Epilog -> {Red,
      
   AbsolutePointSize[3],
      
   Point[{x, y} /. maxX[[2]]],
      
   Point[{x, y} /. maxY[[2]]]}]


      enter image description here






      share|improve this answer











      $endgroup$
















        5












        5








        5





        $begingroup$

        Rationalize the constraint:



        constraint2 = ((x == 0 && 1. <= y <= 12720.) || (1. <= x <= 10712. && 
        
        0 <= y < 2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) + 
        
          2.8484*10^-1 Sqrt[
        
            4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2]) || (10713. <= x <= 
        
         19762. && 
        
        2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) - 
        
          2.8484*10^-1 Sqrt[4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2] < y <
        
          2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) + 
        
          2.8484*10^-1 Sqrt[4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2])) // 
        
    Rationalize[#, 0] & // Simplify;


        With the Rationalized constraint you can use Maximize:



        maxX = Maximize[{x, constraint2}, {x, y}]
        

        
(* {19762, {x -> 19762, y -> 7287}} *)
        

        
constraint2 /. maxX[[2]]
        

        
(* True *)


        EDIT: To find maximum y



        (maxY = Maximize[{y, constraint2}, {x, y}]) // N


        enter image description here



        To plot the region defined by the constraint:



        reg = ImplicitRegion[constraint2, {x, y}];
        

        
Region[reg,
        
 Frame -> True,
        
 FrameLabel -> (Style[#, 12, Bold] & /@ {x, y}),
        
 Epilog -> {Red,
        
   AbsolutePointSize[3],
        
   Point[{x, y} /. maxX[[2]]],
        
   Point[{x, y} /. maxY[[2]]]}]


        enter image description here






        share|improve this answer











        $endgroup$



        Rationalize the constraint:



        constraint2 = ((x == 0 && 1. <= y <= 12720.) || (1. <= x <= 10712. && 
        
        0 <= y < 2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) + 
        
          2.8484*10^-1 Sqrt[
        
            4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2]) || (10713. <= x <= 
        
         19762. && 
        
        2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) - 
        
          2.8484*10^-1 Sqrt[4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2] < y <
        
          2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) + 
        
          2.8484*10^-1 Sqrt[4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2])) // 
        
    Rationalize[#, 0] & // Simplify;


        With the Rationalized constraint you can use Maximize:



        maxX = Maximize[{x, constraint2}, {x, y}]
        

        
(* {19762, {x -> 19762, y -> 7287}} *)
        

        
constraint2 /. maxX[[2]]
        

        
(* True *)


        EDIT: To find maximum y



        (maxY = Maximize[{y, constraint2}, {x, y}]) // N


        enter image description here



        To plot the region defined by the constraint:



        reg = ImplicitRegion[constraint2, {x, y}];
        

        
Region[reg,
        
 Frame -> True,
        
 FrameLabel -> (Style[#, 12, Bold] & /@ {x, y}),
        
 Epilog -> {Red,
        
   AbsolutePointSize[3],
        
   Point[{x, y} /. maxX[[2]]],
        
   Point[{x, y} /. maxY[[2]]]}]


        enter image description here







        share|improve this answer














        share|improve this answer



        share|improve this answer








        edited 10 hours ago

























        answered 10 hours ago









        Bob HanlonBob Hanlon

        61.4k33598




        61.4k33598























            3












            $begingroup$

            You have numbers spread a wide range of magnitudes for no good reason. This range is probably too wide for machine precision arithmetic. Also telling NMinimize explicitly that this an integer optimization problem seems to help. Try this:



            constraint2 = ((x == 0 && 1. <= y <= 12720.) || (1. <= x <= 10712. && 
            
      0 <= y < 
            
       2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) + 
            
        2.8484*10^-1 Sqrt[
            
          4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2]) || (10713. <= 
            
       x <= 19762. && 
            
      2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) - 
            
        2.8484*10^-1 Sqrt[
            
          4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2] < y < 
            
       2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) + 
            
        2.8484*10^-1 Sqrt[
            
          4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2])) // Expand
            

            
maxX = NMaximize[{x, constraint2}, {x, y}, Integers, 
            
  MaxIterations -> 10000]



            {19762., {x -> 19762, y -> 7311}}




            And with your definition of constraint:



            constraint /. maxX[[2]]



            True







            share|improve this answer











            $endgroup$













            • $begingroup$
              But constraint /. x -> 19762 /. y -> 8647 results in False?
              $endgroup$
              – JimB
              11 hours ago










            • $begingroup$
              @JimB D'oh. Yeah, I did the simplification wrong. -.- Thanks for pointing that out.
              $endgroup$
              – Henrik Schumacher
              11 hours ago












            • $begingroup$
              @HenrikSchumacher, thank you for this. This works for x but when I try to find the maximum y similarly, I still get the same message - NMaximize[{y, res}, {x, y}, Integers, MaxIterations -> 100000]. Output: NMaximize::cvdiv: Failed to converge to a solution. The function may be unbounded.
              $endgroup$
              – gaganso
              11 hours ago
















            3












            $begingroup$

            You have numbers spread a wide range of magnitudes for no good reason. This range is probably too wide for machine precision arithmetic. Also telling NMinimize explicitly that this an integer optimization problem seems to help. Try this:



            constraint2 = ((x == 0 && 1. <= y <= 12720.) || (1. <= x <= 10712. && 
            
      0 <= y < 
            
       2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) + 
            
        2.8484*10^-1 Sqrt[
            
          4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2]) || (10713. <= 
            
       x <= 19762. && 
            
      2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) - 
            
        2.8484*10^-1 Sqrt[
            
          4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2] < y < 
            
       2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) + 
            
        2.8484*10^-1 Sqrt[
            
          4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2])) // Expand
            

            
maxX = NMaximize[{x, constraint2}, {x, y}, Integers, 
            
  MaxIterations -> 10000]



            {19762., {x -> 19762, y -> 7311}}




            And with your definition of constraint:



            constraint /. maxX[[2]]



            True







            share|improve this answer











            $endgroup$













            • $begingroup$
              But constraint /. x -> 19762 /. y -> 8647 results in False?
              $endgroup$
              – JimB
              11 hours ago










            • $begingroup$
              @JimB D'oh. Yeah, I did the simplification wrong. -.- Thanks for pointing that out.
              $endgroup$
              – Henrik Schumacher
              11 hours ago












            • $begingroup$
              @HenrikSchumacher, thank you for this. This works for x but when I try to find the maximum y similarly, I still get the same message - NMaximize[{y, res}, {x, y}, Integers, MaxIterations -> 100000]. Output: NMaximize::cvdiv: Failed to converge to a solution. The function may be unbounded.
              $endgroup$
              – gaganso
              11 hours ago














            3












            3








            3





            $begingroup$

            You have numbers spread a wide range of magnitudes for no good reason. This range is probably too wide for machine precision arithmetic. Also telling NMinimize explicitly that this an integer optimization problem seems to help. Try this:



            constraint2 = ((x == 0 && 1. <= y <= 12720.) || (1. <= x <= 10712. && 
            
      0 <= y < 
            
       2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) + 
            
        2.8484*10^-1 Sqrt[
            
          4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2]) || (10713. <= 
            
       x <= 19762. && 
            
      2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) - 
            
        2.8484*10^-1 Sqrt[
            
          4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2] < y < 
            
       2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) + 
            
        2.8484*10^-1 Sqrt[
            
          4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2])) // Expand
            

            
maxX = NMaximize[{x, constraint2}, {x, y}, Integers, 
            
  MaxIterations -> 10000]



            {19762., {x -> 19762, y -> 7311}}




            And with your definition of constraint:



            constraint /. maxX[[2]]



            True







            share|improve this answer











            $endgroup$



            You have numbers spread a wide range of magnitudes for no good reason. This range is probably too wide for machine precision arithmetic. Also telling NMinimize explicitly that this an integer optimization problem seems to help. Try this:



            constraint2 = ((x == 0 && 1. <= y <= 12720.) || (1. <= x <= 10712. && 
            
      0 <= y < 
            
       2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) + 
            
        2.8484*10^-1 Sqrt[
            
          4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2]) || (10713. <= 
            
       x <= 19762. && 
            
      2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) - 
            
        2.8484*10^-1 Sqrt[
            
          4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2] < y < 
            
       2.08565*10^-36 (3.04959*10^39 + 2.24751*10^34 x) + 
            
        2.8484*10^-1 Sqrt[
            
          4.98614*10^8 + 4.65469*10^4 x - 3.63201 x^2])) // Expand
            

            
maxX = NMaximize[{x, constraint2}, {x, y}, Integers, 
            
  MaxIterations -> 10000]



            {19762., {x -> 19762, y -> 7311}}




            And with your definition of constraint:



            constraint /. maxX[[2]]



            True








            share|improve this answer














            share|improve this answer



            share|improve this answer








            edited 11 hours ago

























            answered 11 hours ago









            Henrik SchumacherHenrik Schumacher

            59.4k582165




            59.4k582165












            • $begingroup$
              But constraint /. x -> 19762 /. y -> 8647 results in False?
              $endgroup$
              – JimB
              11 hours ago










            • $begingroup$
              @JimB D'oh. Yeah, I did the simplification wrong. -.- Thanks for pointing that out.
              $endgroup$
              – Henrik Schumacher
              11 hours ago












            • $begingroup$
              @HenrikSchumacher, thank you for this. This works for x but when I try to find the maximum y similarly, I still get the same message - NMaximize[{y, res}, {x, y}, Integers, MaxIterations -> 100000]. Output: NMaximize::cvdiv: Failed to converge to a solution. The function may be unbounded.
              $endgroup$
              – gaganso
              11 hours ago


















            • $begingroup$
              But constraint /. x -> 19762 /. y -> 8647 results in False?
              $endgroup$
              – JimB
              11 hours ago










            • $begingroup$
              @JimB D'oh. Yeah, I did the simplification wrong. -.- Thanks for pointing that out.
              $endgroup$
              – Henrik Schumacher
              11 hours ago












            • $begingroup$
              @HenrikSchumacher, thank you for this. This works for x but when I try to find the maximum y similarly, I still get the same message - NMaximize[{y, res}, {x, y}, Integers, MaxIterations -> 100000]. Output: NMaximize::cvdiv: Failed to converge to a solution. The function may be unbounded.
              $endgroup$
              – gaganso
              11 hours ago
















            $begingroup$
            But constraint /. x -> 19762 /. y -> 8647 results in False?
            $endgroup$
            – JimB
            11 hours ago




            $begingroup$
            But constraint /. x -> 19762 /. y -> 8647 results in False?
            $endgroup$
            – JimB
            11 hours ago












            $begingroup$
            @JimB D'oh. Yeah, I did the simplification wrong. -.- Thanks for pointing that out.
            $endgroup$
            – Henrik Schumacher
            11 hours ago






            $begingroup$
            @JimB D'oh. Yeah, I did the simplification wrong. -.- Thanks for pointing that out.
            $endgroup$
            – Henrik Schumacher
            11 hours ago














            $begingroup$
            @HenrikSchumacher, thank you for this. This works for x but when I try to find the maximum y similarly, I still get the same message - NMaximize[{y, res}, {x, y}, Integers, MaxIterations -> 100000]. Output: NMaximize::cvdiv: Failed to converge to a solution. The function may be unbounded.
            $endgroup$
            – gaganso
            11 hours ago




            $begingroup$
            @HenrikSchumacher, thank you for this. This works for x but when I try to find the maximum y similarly, I still get the same message - NMaximize[{y, res}, {x, y}, Integers, MaxIterations -> 100000]. Output: NMaximize::cvdiv: Failed to converge to a solution. The function may be unbounded.
            $endgroup$
            – gaganso
            11 hours ago


















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